Unconditional nonlinear exponential stability of the motionless conduction-diffusion solution

Nonlinear stability of the motionless state of a heterogeneous fluid with c onstant temperature-gradient and concentration-gradient is studied for both cases of stress-free and rigid boundary conditions. By introducing new ene rgy functionals we have shown that for tau = P-C/P-T less than or equal to 1, <(alpha)over cap> = C/R greater than or equal to 1 the motionless state is always stable and for tau less than or equal to 1, <(alpha)over cap> < 1 the sufficient and necessary conditions for stability coincide, where P-C, P-T, C and R are the Schmidt number, Prandtl number, Rayleigh number for s olute and heat, respectively. Moreover, the criteria guarantees the exponen tial stability.